Population Pharmacokinetic-Pharmacodynamic Modeling: When Modeling Meets Reality


The pharmacokinetic and pharmacodynamic behaviors of many therapeutic agents have inherent complexities that require specialized modeling approaches in order to develop reliable, unbiased models.  Several commonly encountered cases will be presented, including data that has a preponderance of zero values, subjects whose pharmacodynamic trajectories are not predictable based on a structural model, and evaluating models in adaptive dosing situations.

Preponderance of Zero values

For some agents, such as engineered antibodies, it is not unusual to see high specificity and affinity for the target receptors.  The concentration response relationship for these agents is very steep. Thus, such agents can, at therapeutic doses, saturate these receptors.  In this situation, free receptor levels fall to zero and may stay there for protracted periods of time, leading to a large number of zero values in the database.  Owing to the assay methodology used to evaluate receptor occupancy (usually FACS), the zero values are real observations and cannot be deleted from the database.  The distribution of observations is heavy at, and near, the boundary and simple transformation (e.g. converting data to % saturation for instance) does not alleviate this problem.  To model such data, a two part model (truncated delta lognormal distribution) may be utilized   Zero observations are treated in an altered zero fashion and are modeled as discreet, so that the probability of a zero response is modeled.  The natural logarithms of responses with values larger than or equal to 0.1 are modeled as if they arise from a truncated log normal distribution.  This approach provides a distinct advantage because the output is a probability curve showing probability of saturation versus drug concentration, which can be used to guide dose regimens that have optimal receptor saturation.

Models for Indescribable Subpopulations

Most mixture models identify subpopulations based on bimodal or multimodal distributions of eta values.  These subpopulations have an implied association with a missing covariate.  In pharmacodynamic models, such subpopulations may arise due to subjects that are poor responders to a specific treatment. In some cases, subject response may be compromised due to lack of receptors modulating disease, an underlying disease etiology that is different than other patients, or the measured pharmacodynamic marker may be affected due to other events resulting in large, erratic excursions of the marker.  These subjects exhibit a large residue random walk around baseline or around a placebo trajectory.  While on the average they are non-responders, locally in terms of time these subjects might be hyper-responders.

Eta distributions in this latter situation are often badly skewed but not modal, and transformation is not beneficial.  Models are characterized by high residual variability.  Stuart Beal proposed an alternative mixture model which he referred to as the “indescribable model”.  In this unique mixture model, the mixture is based on application of separate functions to a subpopulation where the indescribable population is allowed to progress as an untreated patient might.  This latter mixture model also separates out subjects who are non responders, and can provide valuable diagnostic information into the pharmacological behavior of the drug.

Evaluation of models when adaptive dosing is used

Many therapeutic agents have adaptive dosing strategies which are based on a pharmacodynamic marker of interest.  For example, alemtuzumab is targeted against CD52 which is present on lymphocytes, and acts to reduce their number.  Dosage adjustments are therefore made based on absolute neutrophil count.  Similarly antihypertensive agents are dose adjusted based on measured blood pressure.  When pharmacokinetic/dynamic models for agents that use adaptive dosing strategies are developed, the data used to develop the model includes the adaptive dosing.  However when predictive checks are conducted with such agents, an adaptive dosing strategy must also be implemented.  Failure to account for adaptive dosing in predictive checks can result in biased or inflated prediction intervals because subjects in the simulated data will have undergone dose adjustments for no viable reason.  This impacts particularly for pharmacodynamic models, but also affects pharmacokinetic models for biological agents when the pharmacokinetics are affected by the pharmacodynamic response.